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Strong uniform continuity. (English) Zbl 1161.54003
Strong local continuity is a relative concept: if $$f:X\to Y$$ is a continuous map between metric spaces then not only is $$f\upharpoonright K$$ uniformly continuous whenever $$K$$ is compact: the $$\delta>0$$ that corresponds to the given $$\epsilon>0$$ satisfies the implication “if $$d(x,y)<\delta$$ then $$d(f(x),f(y))<\epsilon$$” even when just one of $$x$$ and $$y$$ belongs to $$K$$. This state of affairs is abbreviated as: $$f$$ is strongly uniformly continuous on $$K$$. The authors study this concept in some depth. They compare the families $$\mathcal{B}^f=\{B:f\upharpoonright B$$ is uniformly continuous$$\}$$ and $$\mathcal{B}_f=\{B:f$$ is strongly uniformly continuous on $$B\}$$; the latter is an ideal (and a bornology if $$f$$ is continuous), the former need not be.
In the second part of the paper the attention shifts to function space topologies; for a bornology $$\mathcal{B}$$ the authors study the topology of strong uniform convergence on members of $$\mathcal{B}$$ (derived from a uniformity wherein closeness of functions is required on neighbourhoods of members of $$\mathcal{B}$$).
Reviewer: K. P. Hart (Delft)

MSC:
 54C05 Continuous maps 54C10 Special maps on topological spaces (open, closed, perfect, etc.) 54C35 Function spaces in general topology 54E15 Uniform structures and generalizations
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